Elements of Electromagnetism (I)

Electromagnetism is really cute. In this post (and its sequel) my aim is to provide a very broad overview of what physicist mean when they talk about the theory of electromagnetism or Maxwell theory. First of all I would like to list some of the references I used to get my information from and some other documents were one can pursue further details.

1. “Gauge theories in particle physics” by Aitchison and Hey – chapter 2 provides most of what I mention here.
2. “Classical Electrodynamics” by JDJ – Just in case people really wanted to learn electromagnetism.
3. “The conceptual basis of quantum field theory” by G’tH – Look here for more details on Yang-Mills theory, a generalization of Maxwell theory.
4. “Special Relativity: from Einstein to strings” by Schwarz and Schwarz – Here are some other topics I wanted to talk about but maybe will not have time to cover.
5. “A first course in string theory” by BZ – Stuff on Kalb-Ramond and Born-Infeld.

So I was requested to talk about electromagnetism as a gauge theory. What is electromagnetism? What is a gauge theory? And how is electromagnetism a gauge theory? For now I will start with the first question. We will work our way through some history and we will reach a point where we will say something like “… in this sense E&M is a gauge theory of local phase transformations… “. This will hopefully answer the third question, leaving the second question (what is a gauge theory) for another occasion.

So what is electromagnetism? My answer is that it depends who you ask.

If you ask a four-year old he/she might respond that probably it has something to do with electrical wires and magnets. That is true. As a physicist one can define electromagnetism as all the physical phenomena that is described by the electric and the magnetic fields.

Later you ask a college freshmen. He/she will probably recite Maxwell’s equations:

     

for the electric field and also

     

for the magnetic field . I have used the symbol , which means the flux of the vector field through the two-dimensional surface . Also, is a closed surface, while means the boundary of an open surface. By vector field I mean a real function that assigns a 3-dimensional vector at each point in space and at each instant in time. This set of equations is also know as the “integral version” of Maxwell’s equations.

You wait some years now and return to find a college junior working with another set of equations:

     

     

These are not really that mysterious: one can obtain them by starting from the freshmen equations and applying theorems from vector calculus, like the divergence and the curl theorems. This set is called the “differential” or the “awesome” version of Maxwell’s equations.

So far I have mentioned nothing about the sources for the electric and magnetic field. Roughly speaking, the electric field is sourced by electric charges which are point-like. One can construct a solid charged body by adding infinitesimal charges. On the other hand, the magnetic field is sourced by moving charges. A collection of moving charges is called a current . This business with moving charges may remind you of flow. Indeed, physicist constantly invoke the principle of charge conservation: charge cannot be created or destroyed at a given point in space. While shifting from the integral to the differential version I have introduced the density of charge per unit volume and the density of current per unit area . The vector character of describe the direction of current flow. It turns out that the principle of charge conservation can be described by a continuity equation:

The integral form of Maxwell’s equations sometimes are easier to apply to basic problems (like the field of a point charge or the field of a long wire) than the differential form (no puns intended). From now on we will be working with the differential version, which allows a straightforward derivation of other features of classical electromagnetism.

For instance, the fact that the magnetic field has vanishing divergence implies that it can be written as the curl of some other vector field:

This in turn allows us to say that

The functions and are known as the scalar and vector potentials, respectively. Wait a second! Now I am saying that both the electric and the magnetic field can be expressed in terms of some other functions. Why aren’t these functions THE fundamental objects? It turns out that the potentials are not very physical. Why is this? Because one can obtain the same electric and magnetic fields from different sets of potentials. For example, since the magnetic field is the curl of something, we can add a gradient to this something and still obtain the same potential, I.E.

   and      give the same

This only works for smooth functions of course. If we make this change to the vector potential, we also notice that

and    give the same

We say that Maxwell’s equations are left invariant (unchanged) under this type of transformation. This will later be called a gauge transformation with the function called the gauge parameter. The fact that many potentials give the same fields is the reason why the potentials are not fundamental.

Let us now consider the case with no sources. Physicist call this the vacuum. It turns out that electric and magnetic fields can still exist in such a case. This is the reason why the fields are the fundamental objects and not the sources. In the vacuum we have and also . Then the set of Maxwells take the form

     

     

This certainly looks more symmetric. In fact, if there were magnetic charges, we will have even more symmetry. But physicist have not been able to find magnetic charges in nature so far, besides the Valentine’s Day Monopole. Going back to physics, one can manipulate the Maxwells to obtain the following expressions:

   and   .

This is nothing but the wave equation for a wave with (constant!!!!) velocity . This is the reason why I kept explicit values of the constants and . At first this was not that big. But when people in the late 19th century tried to combine this result with Galilean physics one obtains some inconsistencies. It was Albert Einstein who was able to understand what was going on: electromagnetic disturbances always move with the same speed and space and time sort of stretch or shrink in order to keep this true. This is nothing but the foundation of Special Relativity. From now on I will use units such that .

But before we turn into relativity, let us mention one last thing about the wave equations. If one writes down the wave equations in terms of the potentials one can see another reason why the potentials are not fundamental: after a gauge transformation, the wave equation for the potential changes. Then one can imagine different choices of the gauge parameter corresponding to different measuring setups. One setup will have some differential equation for the potentials, another setup some other equation and so on. But, since the fields are gauge-invariant, the different gauge-observers agree on the same behavior of the electric and magnetic fields.

After all this you think to yourself: “This is getting very interesting!”. Indeed, if one bugs a college senior, he/she will be using Maxwell’s equations in a funky, covariant form. But let us stop now to catch our breath and continue later.

Tags: Electromagnetism